The Klein View of Geometry

Fundamental Theorems

In euclidean geometry, for two points A and A', there are elements of E(2) which map A to A'.
A translation is one possibility. In our other geometries, a similar result holds. In the similarity
and inversive geometries, this is easy to see. In hyperbolic geometry, it is a consequence of
the Interchange Lemma.

However, in euclidean geometry, if L = (A,B) and L' = (A',B') are lists of distinct points, then
we can map L to L' if and only if |AB| =|A'B'|.

In similarity geometry, we can map pairs to pairs. We shall call this

The Fundamental Theorem of Similarity Geometry
If L = (A,B) and L' = (A',B') are lists of distinct points of E, then
there is a unique element of S+(2) which maps L to L'.

Proof of the theorem

We can go no further since similarity geometry as the property of angle. Thus we can map
(A,B,C) to (A',B',C') if and only if the three angles defined by each list are equal.

In inversive geometry, the situation is more interesting. We have

The Fundamental Theorem of Inversive Geometry

If L = (A,B,C) and L' = (A',B',C') are lists of distinct points of E+, then
there is a unique element of I+(2) which maps L to L'.

This result and its proof appear on the inversive group pages.

In hyperbolic geometry, we have the property of hyperbolic distance, so the situation is similar
to that in euclidean geometry.

The other geometries we consider, affine and projective, have the property that two distinct points
lie on a unique line. Also, the group elements map lines to lines, so we cannot hope to map three
collinear points to three non-collinear points, or vice versa. With this restriction, we do have
Fundamental Theorems for these geometries.

The Fundamental Theorem of Affine Geometry

If L = (A,B,C) and L' = (A',B',C') are lists of non-collinear points of E, then
there is a unique element of A(2) which maps L to L'.

A proof can be found in the affine group pages.

Since affine transformations preserve ratio along a line, we cannot in general map
three collinear points to another three collinear points.

The Fundamental Theorem of Projective Geometry

If L = (A,B,C,D) and L' = (A',B',C',D) are lists of points of RP2, each with no three collinear, then
there is a unique element of P(2) which maps L to L'.

A proof can be found in the projective group pages.

Since projective transformations preserve the cross-ratio of four collinear p-points, we cannot
in general map four collinear points of RP2 to another four collinear points.

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